Results 1 
7 of
7
Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
Abstract

Cited by 68 (4 self)
 Add to MetaCart
We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
An effective decision procedure for linear arithmetic with integer and real variables
 ACM Transactions on Computational Logic (TOCL
, 2005
"... This article considers finiteautomatabased algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. T ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
This article considers finiteautomatabased algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. The contribution of this article is to show, using topological arguments, that only a restricted class of automata on infinite words are necessary for handling real and integer linear arithmetic. This allows the use of substantially simpler algorithms, which have been successfully implemented.
On the Expressiveness of Real and Integer Arithmetic Automata (Extended Abstract)
 ICALP'98, LNCS 1443
, 1998
"... If read digit by digit, a ndimensional vector of integers represented in base r can be viewed as a word over the alphabet r n . It has been known for some time that, under this encoding, the sets of integer vectors recognizable by finite automata are exactly those de nable in Presburger arithmetic ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
If read digit by digit, a ndimensional vector of integers represented in base r can be viewed as a word over the alphabet r n . It has been known for some time that, under this encoding, the sets of integer vectors recognizable by finite automata are exactly those de nable in Presburger arithmetic if independence with respect to the base is required, and those de nable in a slight extension of Presburger arithmetic if only a specific base is considered. Using the same encoding idea, but moving to infinite words, finite automata on infinite words can recognize sets of real vectors. This leads to the question of which sets of real vectors are recognizable by finite automata, which is the topic of this paper. We show that the recognizable sets of real vectors are those definable in the theory of reals and integers with addition and order, extended with a special basedependent predicate that tests the value of a specified digit of a number. Furthermore, in the course of proving that sets of vectors de ned in this theory are recognizable by finite automata, we show that linear equations and inequations have surprisingly compact representations by automata, which leads us to believe that automata accepting sets of real vectors can be of more than theoretical interest.
A Generalization of Cobham’s Theorem to Automata over Real Numbers
, 2008
"... This article studies the expressive power of finitestate automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the firstorder additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi auto ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
This article studies the expressive power of finitestate automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the firstorder additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi automata, regardless of the encoding base r> 1. In this article, we prove the reciprocal property, i.e., that a subset of R that is recognizable by weak deterministic automata in every base r> 1 is necessarily definable in 〈R, Z, +, <〉. This result generalizes to real numbers the wellknown Cobham’s theorem on the finitestate recognizability of sets of integers. Our proof gives interesting insight into the internal structure of automata recognizing sets of real numbers, which may lead to efficient data structures for handling these sets.
A Survey of Arithmetical Definability
, 2002
"... We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.
A Generalization of Semenov’s Theorem to Automata over Real Numbers
"... This work studies the properties of finite automata recognizing vectors with real components, encoded positionally in a given integer numeration base. Such automata are used, in particular, as symbolic data structures for representing sets definable in the firstorder theory 〈R, Z, +, ≤〉, i.e., the ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This work studies the properties of finite automata recognizing vectors with real components, encoded positionally in a given integer numeration base. Such automata are used, in particular, as symbolic data structures for representing sets definable in the firstorder theory 〈R, Z, +, ≤〉, i.e., the mixed additive arithmetic of integer and real variables. They also lead to a simple decision procedure for this arithmetic. In previous work, it has been established that the sets definable in 〈R, Z, +, ≤ 〉 can be handled by a restricted form of infiniteword automata, weak deterministic ones, regardless of the chosen numeration base. In this paper, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable in 〈R, Z, +, ≤〉. This result can be seen as a generalization to the mixed integer and real domain of Semenov’s theorem, which characterizes the sets of integer vectors recognizable by finite automata in multiple bases. It also extends to multidimensional vectors a similar property recently established for sets of numbers. As an additional contribution, the techniques used for obtaining our main result lead to valuable insight into the internal structure of automata recognizing sets of vectors definable in 〈R, Z, +, ≤〉. This structure might be exploited in order to improve the efficiency of representation systems and decision procedures for this arithmetic.
DOI: 10.1007/9783642029592_34 A Generalization of Semenov’s Theorem to Automata over Real Numbers ⋆
, 2009
"... Abstract This work studies the properties of finite automata recognizing vectors with real components, encoded positionally in a given integer numeration base. Such automata are used, in particular, as symbolic data structures for representing sets definable in the firstorder theory 〈R, Z, +, ≤〉, i ..."
Abstract
 Add to MetaCart
Abstract This work studies the properties of finite automata recognizing vectors with real components, encoded positionally in a given integer numeration base. Such automata are used, in particular, as symbolic data structures for representing sets definable in the firstorder theory 〈R, Z, +, ≤〉, i.e., the mixed additive arithmetic of integer and real variables. They also lead to a simple decision procedure for this arithmetic. In previous work, it has been established that the sets definable in 〈R, Z, +, ≤ 〉 can be handled by a restricted form of infiniteword automata, weak deterministic ones, regardless of the chosen numeration base. In this paper, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable in 〈R, Z, +, ≤〉. This result can be seen as a generalization to the mixed integer and real domain of Semenov’s theorem, which characterizes the sets of integer vectors recognizable by finite automata in multiple bases.